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"... Abstract. We consider a cyclic approach to inductive reasoning in the setting of first-order logic with inductive definitions. We present a proof system for this language in which proofs are represented as finite, locally sound derivation trees with a “repeat function ” identifying cyclic proof sect ..."

Abstract. We consider a cyclic approach to inductive reasoning in the setting of first-order logic with inductive definitions. We present a proof system for this language in which proofs are represented as finite, locally sound derivation trees with a “repeat function ” identifying cyclic proof sections. Soundness is guaranteed by a well-foundedness condition formulated globally in terms of traces over the proof tree, following an idea due to Sprenger and Dam. However, in contrast to their work, our proof system does not require an extension of logical syntax by ordinal variables. A fundamental question in our setting is the strength of the cyclic proof system compared to the more familiar use of a non-cyclic proof system using explicit induction rules. We show that the cyclic proof system subsumes the use of explicit induction rules. In addition, we provide machinery for manipulating and analysing the structure of cyclic proofs, based primarily on viewing them as generating regular infinite trees, and also formulate a finitary trace condition sufficient (but not necessary) for soundness, that is computationally and combinatorially simpler than the general trace condition. 1

... [19] and in program verification based on automata [20]. It has also been studied in the context of tableau-style proof systems for the µ-calculus by Sprenger and Dam [17, 18] and Schöpp and Simpson =-=[15]-=-, following an approach proposed by Dam and Gurov [5]. Our aim is to study cyclic reasoning in the relatively simple, yet expressive context of first-order logic extended with ordinary inductive defin...

"... In this paper we study induction in the context of the first-order µ-calculus with explicit approximations. We present and compare two Gentzen-style proof systems each using a different type of induction. The first is ..."

In this paper we study induction in the context of the first-order µ-calculus with explicit approximations. We present and compare two Gentzen-style proof systems each using a different type of induction. The first is

"... We propose a novel approach to proving the termination of heapmanipulating programs, which combines separation logic with cyclic proof within a Hoare-style proof system. Judgements in this system express (guaranteed) termination of the program when started from a given line in the program and in a s ..."

We propose a novel approach to proving the termination of heapmanipulating programs, which combines separation logic with cyclic proof within a Hoare-style proof system. Judgements in this system express (guaranteed) termination of the program when started from a given line in the program and in a state satisfying a given precondition, which is expressed as a formula of separation logic. The proof rules of our system are of two types: logical rules that operate on preconditions; and symbolic execution rules that capture the effect of executing program commands. Our logical preconditions employ inductively defined predicates to describe heap properties, and proofs in our system are cyclic proofs: cyclic derivations in which some inductive predicate is unfolded infinitely often along every infinite path, thus allowing us to discard all infinite paths in the proof by an infinite descent argument. Moreover, the use of this soundness condition enables us to avoid the explicit construction and use of ranking functions for termination. We also give a completeness result for our system, which is relative in that it relies upon completeness of a proof system for logical implications in separation logic. We give examples illustrating our approach, including one example for which the corresponding ranking function is non-obvious: termination of the classical algorithm for in-place reversal of a (possibly cyclic) linked list.

...uces an infinite descending sequence of values from a well-ordered set (Lee et al. 2001). Here, we employ the existing techniques of cyclic proof (Brotherston 2007, 2006, 2005; Sprenger and Dam 2003; =-=Schöpp and Simpson 2002-=-) to consider and then dismiss potentially infinite computations in the manner described above. A cyclic pre-proof in our system is formed from partial derivation trees by identifying every ‘bud’ (a n...

"... We investigate a Gentzen-style proof system for the first-order µ-calculus based on cyclic proofs, produced by unfolding fixed point formulas and detecting repeated proof goals. Our system uses explicit ordinal variables and approximations to support a simple semantic induction discharge conditio ..."

We investigate a Gentzen-style proof system for the first-order µ-calculus based on cyclic proofs, produced by unfolding fixed point formulas and detecting repeated proof goals. Our system uses explicit ordinal variables and approximations to support a simple semantic induction discharge condition which ensures the well-foundedness of inductive reasoning. As the main result of this paper we propose a new syntactic discharge condition based on traces and establish its equivalence with the semantical condition. We give an automata-theoretic reformulation of this condition which is more suitable for practical proofs. For a detailed

...nite descent arguments. Recently, tableau-style proof systems for the µ-calculus employing cyclic proofs were developed first by Dam and Gurov [19] and then investigated further by Schöpp and Simpson =-=[58]-=- and by Sprenger and Dam [62, 63]. These systems embody an infinite descent principle for the µ-calculus, based upon an ordinal indexing of the approximants by which the least and greatest fixed point...